numbers-3000.2.0.2: Various number types

Data.Number.Dif

Description

The Dif module contains a data type, Dif, that allows for automatic forward differentiation.

All the ideas are from Jerzy Karczmarczuk's work, see http://users.info.unicaen.fr/~karczma/arpap/diffalg.pdf.

A simple example, if we define

foo x = x*x

then the function

foo' = deriv foo

will behave as if its body was 2*x.

Synopsis

# Documentation

data Dif a Source #

The Dif type is the type of differentiable numbers. It's an instance of all the usual numeric classes. The computed derivative of a function is is correct except where the function is discontinuous, at these points the derivative should be a Dirac pulse, but it isn't.

The Dif numbers are printed with a trailing ~~ to indicate that there is a "tail" of derivatives.

Instances

 Eq a => Eq (Dif a) Source # Methods(==) :: Dif a -> Dif a -> Bool #(/=) :: Dif a -> Dif a -> Bool # (Floating a, Eq a) => Floating (Dif a) Source # Methodspi :: Dif a #exp :: Dif a -> Dif a #log :: Dif a -> Dif a #sqrt :: Dif a -> Dif a #(**) :: Dif a -> Dif a -> Dif a #logBase :: Dif a -> Dif a -> Dif a #sin :: Dif a -> Dif a #cos :: Dif a -> Dif a #tan :: Dif a -> Dif a #asin :: Dif a -> Dif a #acos :: Dif a -> Dif a #atan :: Dif a -> Dif a #sinh :: Dif a -> Dif a #cosh :: Dif a -> Dif a #tanh :: Dif a -> Dif a #asinh :: Dif a -> Dif a #acosh :: Dif a -> Dif a #atanh :: Dif a -> Dif a #log1p :: Dif a -> Dif a #expm1 :: Dif a -> Dif a #log1pexp :: Dif a -> Dif a #log1mexp :: Dif a -> Dif a # (Fractional a, Eq a) => Fractional (Dif a) Source # Methods(/) :: Dif a -> Dif a -> Dif a #recip :: Dif a -> Dif a # (Num a, Eq a) => Num (Dif a) Source # Methods(+) :: Dif a -> Dif a -> Dif a #(-) :: Dif a -> Dif a -> Dif a #(*) :: Dif a -> Dif a -> Dif a #negate :: Dif a -> Dif a #abs :: Dif a -> Dif a #signum :: Dif a -> Dif a # Ord a => Ord (Dif a) Source # Methodscompare :: Dif a -> Dif a -> Ordering #(<) :: Dif a -> Dif a -> Bool #(<=) :: Dif a -> Dif a -> Bool #(>) :: Dif a -> Dif a -> Bool #(>=) :: Dif a -> Dif a -> Bool #max :: Dif a -> Dif a -> Dif a #min :: Dif a -> Dif a -> Dif a # Read a => Read (Dif a) Source # MethodsreadsPrec :: Int -> ReadS (Dif a) #readList :: ReadS [Dif a] #readPrec :: ReadPrec (Dif a) # Real a => Real (Dif a) Source # MethodstoRational :: Dif a -> Rational # RealFloat a => RealFloat (Dif a) Source # MethodsfloatRadix :: Dif a -> Integer #floatDigits :: Dif a -> Int #floatRange :: Dif a -> (Int, Int) #decodeFloat :: Dif a -> (Integer, Int) #encodeFloat :: Integer -> Int -> Dif a #exponent :: Dif a -> Int #significand :: Dif a -> Dif a #scaleFloat :: Int -> Dif a -> Dif a #isNaN :: Dif a -> Bool #isInfinite :: Dif a -> Bool #isDenormalized :: Dif a -> Bool #isNegativeZero :: Dif a -> Bool #isIEEE :: Dif a -> Bool #atan2 :: Dif a -> Dif a -> Dif a # RealFrac a => RealFrac (Dif a) Source # MethodsproperFraction :: Integral b => Dif a -> (b, Dif a) #truncate :: Integral b => Dif a -> b #round :: Integral b => Dif a -> b #ceiling :: Integral b => Dif a -> b #floor :: Integral b => Dif a -> b # Show a => Show (Dif a) Source # MethodsshowsPrec :: Int -> Dif a -> ShowS #show :: Dif a -> String #showList :: [Dif a] -> ShowS #

val :: Dif a -> a Source #

The val function takes a Dif number back to a normal number, thus forgetting about all the derivatives.

df :: (Num a, Eq a) => Dif a -> Dif a Source #

The df takes a Dif number and returns its first derivative. The function can be iterated to to get higher derivaties.

mkDif :: a -> Dif a -> Dif a Source #

The mkDif takes a value and Dif value and makes a Dif number that has the given value as its normal value, and the Dif number as its derivatives.

dCon :: Num a => a -> Dif a Source #

The dCon function turns a normal number into a Dif number with the same value. Not that numeric literals do not need an explicit conversion due to the normal Haskell overloading of literals.

dVar :: (Num a, Eq a) => a -> Dif a Source #

The dVar function turns a number into a variable number. This is the number with with respect to which the derivaticve is computed.

deriv :: (Num a, Num b, Eq a, Eq b) => (Dif a -> Dif b) -> a -> b Source #

The deriv function is a simple utility to take the derivative of a (single argument) function. It is simply defined as

 deriv f = val . df . f . dVar

unDif :: (Num a, Eq a) => (Dif a -> Dif b) -> a -> b Source #

Convert a Dif function to an ordinary function.